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I have a subset $B\subset\mathbb{R}^n\times\mathbb{R}^m$ that I want to show has measure zero. I know that the sections $B^x = {y : (x,y)\in B}$ all have measure zero. I do not know if $B$ is measurable. Is this enough to conclude that $B$ is a measurable set with measure zero?

EDIT : I would like to say that when randomly selecting $a\in\mathbb{R}^n\times\mathbb{R}^m$, I am almost always not in $B$. Since I know that $B^x = {y : (x,y)\in B}$ all have measure zero, it seems like a reasonable conclusion. I don't know enough probability or measure theory to put this in a rigorous way, so any suggestions would be great.

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# Sections measure zero imply set is measure zero?

I have a subset $B\subset\mathbb{R}^n\times\mathbb{R}^m$ that I want to show has measure zero. I know that the sections $B^x = {y : (x,y)\in B}$ all have measure zero. I do not know if $B$ is measurable. Is this enough to conclude that $B$ is a measurable set with measure zero?